If `f((x)/(y))=(f(x))/(f(y)) forall x, y in R, y ne 0 and f'(x)` exists for all x, `f(2) = 4`. Then, `f(5)` is

Options

A. 3
B. 5
C. 25
D. None of the above

Answer

To solve the problem, we need to analyze the given functional equation and derive the function \( f(x) \).

### Step 1: Analyze the functional equation
We are given:
\[ f\left(\frac{x}{y}\right) = \frac{f(x)}{f(y)} \quad \text{for all } x, y \in \mathbb{R}, y \neq 0 \]
This suggests a relationship between the function values at different points.

### Step 2: Substitute specific values
Let's substitute \( y = 1 \):
\[ f\left(\frac{x}{1}\right) = \frac{f(x)}{f(1)} \implies f(x) = \frac{f(x)}{f(1)} \]
This implies that \( f(1) \) must be 1 (assuming \( f(x) \neq 0 \)).

### Step 3: General form of the function
From the functional equation, we can conjecture that the function might be of the form:
\[ f(x) = x^a \]
for some constant \( a \).

### Step 4: Use the given condition \( f(2) = 4 \)
Substituting \( x = 2 \) into our conjectured form:
\[ f(2) = 2^a = 4 \]
This simplifies to:
\[ 2^a = 2^2 \implies a = 2 \]

### Step 5: Determine the function
Now that we have \( a = 2 \), we can write:
\[ f(x) = x^2 \]

### Step 6: Find \( f(5) \)
Now we need to find \( f(5) \):
\[ f(5) = 5^2 = 25 \]

### Conclusion
Thus, the value of \( f(5) \) is:
\[ \boxed{25} \]